Computational Complexity of Outer-Independent Total and Total Roman Domination Numbers in Trees

An outer-independent total dominating set (OITDS) of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is a set <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> of vertices of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> such that every vertex of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> has a neighbor in <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>, and the set <inline-formula> <tex-math notation="LaTeX">$V(G)\setminus D$ </tex-math></inline-formula> is independent. The outer-independent total domination number of a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\gamma _{oit}(G)$ </tex-math></inline-formula>, is the minimum cardinality of an OITDS of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. An outer-independent total Roman dominating function (OITRDF) on a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is a function <inline-formula> <tex-math notation="LaTeX">$f: V(G) \rightarrow \{0, 1, 2\}$ </tex-math></inline-formula> satisfying the conditions that every vertex <inline-formula> <tex-math notation="LaTeX">$u$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$f(u)=0$ </tex-math></inline-formula> is adjacent to at least one vertex <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$f(v)=2$ </tex-math></inline-formula>, every vertex <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$f(x)\geq 1$ </tex-math></inline-formula> is adjacent to at least one vertex <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$f(y)\geq 1$ </tex-math></inline-formula>, and any two different vertices <inline-formula> <tex-math notation="LaTeX">$a,b$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$f(a)=f(b)=0$ </tex-math></inline-formula> are not adjacent. The minimum weight <inline-formula> <tex-math notation="LaTeX">$\omega (f) =\sum _{v\in V(G)}f(v)$ </tex-math></inline-formula> of any OITRDF <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is the outer-independent total Roman domination number of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\gamma _{oitR}(G)$ </tex-math></inline-formula>. A graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called an outer-independent total Roman graph (OIT-Roman graph) if <inline-formula> <tex-math notation="LaTeX">$\gamma _{oitR}(G)=2\gamma _{oit}(G)$ </tex-math></inline-formula>. In this paper, we propose dynamic programming algorithms to compute the outer-independent total domination number and the outer-independent total Roman domination number of a tree, respectively. Moreover, we characterize all OIT-Roman graphs and give a linear time algorithm for recognizing an OIT-Roman graph.